Question: Simplify and expand the following expression: $ \dfrac{4x}{4x + 2}+\dfrac{3x}{5x - 2} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4x + 2)(5x - 2)$ Multiply the first term by $\dfrac{5x - 2}{5x - 2}$ $ \begin{align*} \dfrac{4x}{4x + 2} \times \dfrac{5x - 2}{5x - 2} & = \dfrac{(4x)(5x - 2)}{(4x + 2)(5x - 2)} \\ & = \dfrac{20x^2 - 8x}{(4x + 2)(5x - 2)}\end{align*} $ Multiply the second term by $\dfrac{4x + 2}{4x + 2}$ $ \begin{align*} \dfrac{3x}{5x - 2} \times \dfrac{4x + 2}{4x + 2} & = \dfrac{(3x)(4x + 2)}{(5x - 2)(4x + 2)} \\ & = \dfrac{12x^2 + 6x}{(5x - 2)(4x + 2)}\end{align*} $ Now we have: $ = \dfrac{20x^2 - 8x}{(4x + 2)(5x - 2)} + \dfrac{12x^2 + 6x}{(5x - 2)(4x + 2)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{20x^2 - 8x + 12x^2 + 6x}{(4x + 2)(5x - 2)} $ $ = \dfrac{32x^2 - 2x}{(4x + 2)(5x - 2)}$ Expand the denominator: $ = \dfrac{32x^2 - 2x}{20x^2 + 2x - 4}$ Simplify: $ = \dfrac{16x^2 - x}{10x^2 + x - 2}$